https://www.cs.princeton.edu/courses/archive/fall13/cos521/lecnotes/lec12.pdf

## lovely Turing mixer theory

## Accumulation Point — from Wolfram MathWorld

http://mathworld.wolfram.com/AccumulationPoint.html

this is the sense of turing- the point at which the measure of diffusion (of colinear differentials in crypto) has become chaotic

recall turing used a doubly transitive operator to mix the plaintext differentials within the output space

## Linear representation theory of quaternion group – Groupprops

https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_quaternion_group

Center and zero

More great Turing

View as CPU design, instruction set for rational – i.e. Probability densities seen in cryptanalysis

View in terms of early comp sci, searching for a computable group (suited to improbability calculus)

## Subgroup structure of quaternion group – Groupprops

https://groupprops.subwiki.org/wiki/Subgroup_structure_of_quaternion_group

Relate center to quotient as Klein.

We saw this in tunny

## Quaternion group picture

http://math.stackexchange.com/questions/1971591/quaternion-group

Contrast with fano plane which holds for octionions

## More Turing simila

Turings on permutation argument

http://math.stackexchange.com/questions/866026/quaternion-group-as-permutation-group

## Eigenvalues and eigenkets in turings on permutations

Looking at turnings overall argument form, I now see that he does address pure states. When he talks about those group element with particular discriminators,these are the eigenkets for his rotational (pre-measurement) operator.

His system is still constrained to be looking at lambda2 (max difference from constants etc).

Interesting also that the inner product norm is to him just a (easy sum of squares) measure of the squiggle path one takes, with respect to another vector (including his eigenket vectors, k).

## Metric and Topological Spaces index

http://www-history.mcs.st-and.ac.uk/~john/MT4522/index.html

excellent uk teaching.

as turing was taught it.

## Fano plane – WikiVisually

http://wikivisually.com/wiki/Fano_plane

Back to the future

Gives a nice notion of distance for measurable probability densities such as those found in Markova ciphers resisting 1943 era differential cryptanalysis

## http://www.ams.org/journals/bull/1931-37-12/S0002-9904-1931-05281-2/S0002-9904-1931-05281-2.pdf

The Turing era teaching on doubly transitive and a sequence of h1….hi, playing the role of n des rounds making a markov cipher

## http://download.springer.com/static/pdf/540/chp%253A10.1007%252F3-540-48285-7_41.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F3-540-48285-7_41&token2=exp=1486607401~acl=%2Fstatic%2Fpdf%2F540%2Fchp%25253A10.1007%25252F3-540-48285-7_41.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Fchapter%252F10.1007%252F3-540-48285-7_41*~hmac=ba58ced97b0e09b59ce3340716e1303b90efc7c0e07176f358265b8de53d6197

Doubly transitive like Turing P22

Simple groups

Now we see why Turing left unstated the relationship to differential cryptanalysis.

We must defend and protect federal networks and data,” Trump said during a meeting on cybersecurity. “We operate these networks on behalf of the American people and they are very important and very sacred.” The executive order had been scheduled for signing after the meeting. It was unclear when it would be signed.

Link needed.

Because NSA objected and , just as with the dhs attempt to usurp NSA role earlier, was able to state why things must remain with NSA. ;and that’s not even discounting the alternatives lack of deployed capability).

The emotional reason doesn’t take a lot of understanding (and even trump – t for torture and p for pussy, recall – understands that without control over the metadata, you cannot spy. And that means spying as much on congress as vlad the impaler.

At this point NSA will be pitching trump on how they validate and protect him (oh king, imperatur) as the preatorians. At his level of paranoia it will be doubly effective.

Let C be a conjugacy class of the symmetric group Sn . Recall that C consists of all of the permutations of a given cycle type. Hence we may define the support of C, denoted by supp(C), to be the number of nonfixed digits under the action of a permutation in C. If C has “large” support, then each element of C“moves” a lot of letters. If the permutations in C are all even, then we say that C is an even conjugacy class. If the permutations in C are all odd, we say that C is an odd conjugacy class. For n≥5, the subgroup generated by C, denoted by C, is Sn if C is odd, and An if C is even.

## tunny run and quantum coins

think of each tunny run as a particular biased quantum coin space.

building now see on each side of e (the changing slit pattern)a torus. 2 donuts in a box

ie the screen is the torus surface

we want the circle packing to measure the embedding into spectral uniformity

we want gap minimzed when number of tangents on bith torus are equal

## Morphogenesus

Now we have broken through on des by applying turings quantum walk (on graphs whose eigenvalues are functions of the characters) we can turn to his 2d lattice, coded way of expressing his cipher knowhow.

What’s is interesting will be to relate light cones and Padic labeling to the n particle case of the des e functions “slit”, as subkey expansion evolve the positions of the slits, and thereby imposes a discriminator (attacking des)

## Shifting coined operators

Turing describes a quantum walk, using only the apparatus of a rotor machine.

What is cute is the very simple way the evolution shifts (the distances between the transformed state).

## Lovely orientation material for understanding math of colossus cryptographic attack on Tunney

## HTTPS://yorkporc.files.wordpress.com/2011/03/image19.png

S box nibble sized graph evolution (with quantum coin from support between pairs of plaintext chars).

Distinguished from e(), that de-localized flows, simulating quantum coherence

## CS359G Lecture 4: Spectral Partitioning | in theory

We even get (after 3 years) why the tunny alphabet was ordered the way it was, when applied to cryptanalysis. See spectral partitioning and the way in which nodes were ordered.

## CS359G Lecture 4: Spectral Partitioning | in theory

As we will see in a later lecture, there is a nearly linear time algorithm that finds a vector {x} for which the expression {\delta} in the lemma is very close to {1-\lambda_2}, so, overall, for any graph {G} we can find a cut of expansion {O(\sqrt {h(G)})} in nearly linear time.

…for any tunny “run” known to exhibit a stats measurable bias, through just counting (on colossus).

Think of each std deviation worth of amplification as refining the cutset And tuning to the ciphertext, to best decide between improbably candidates (wheel s that in x or still in the edge expansion set)

## Des spectral gap theory

So the 7point geometry is special since it embodies cut sets (those nodes on the circle vs the “outer” triangle

U also look at the area of each as probability (of the event space). Noting obviously circle is less than triangle.

This is highly pertinent to quantum walks.

Now in terms of resistance to differential cryptanalysis one has to think of tunny attack: which wanted to chose just that cutest (between circle and triangle) that allowed a decision on two hypothesis tied to the improbability of each.

To resist D.C., the weight difference between the two h must be within the spectral gap.

Do I get the two sides of the des design argument. We are requiring the quantum random walk but also needing it amplifications to evolve the density to within the spectral gap do that h cannot be weighted in an attack.

Ok so I was right on my intuition, the other day. Now we have histologucal and pure math support.

## Turing reasoning when search

## How des works – again

So here we are again, explaining

the principles upon which des is designed. Yes it’s the nth time; and the same caveat as before: go anywhere else to learn someone’s description of the algorithm. Here we motivate the design.

So imagine you are studying the infamous quantum mechanics 2 hole screen – separating electron gun on the left from intensity screen. Or the 1 hole. Or the two holes shifted up (or down) a bit on the grate.

Oh and don’t forget that we can flip the grate, should we want the gun on the right (and the screen on the left).

After all like des, quantum mechanics is inevitably reversible, since “information” is conserved.

Now imagine that the e function of des is the grate.

And 2 of the six input bits of the current sbox are the slits in the grates.

The purpose of the way the e function shift inputs left (and right) is to simulate shifting the slits in the grate ip and down.

Why?

So that the qm normal density is shifted.

Our goal is to uniformly fill an intensity space, on the screen. Or better, average the left and right intensity screen (per the classical functional form.

Now, even though we shift our normal curve up and down (as the number of supports in the pair wise plaintext’s chats induce a key particle to move 1,2 3… lambada from the center, within the normal curve (and while constructing/destructing as we go) we still need the addition of intensity curves to uniformly fill the output space.

And here is where comes in the particular key schedule. It’s particular sequencing moves the grate around not only so feistels multiplexor covers all subpace but does so in such manner that guarantees that the concentration of intensity at any point (in 2d intensity space) is never more than the second eigenvalue gap.

Now recall that des does not have reversible sboxes. But we don’t need them! After all we have interleaved diffraction grates, since left 2 right we have half s des round and (right 2 left) we have the other half.

Now view the des subkey generatio functions own (highly programmed) bit duplication as a means of subtly (at huge granularity) measuring the quantum effect, thus influencing just how the left and right particle motions occur – giving a characteristic.

And ensuring that there exists no matrix representation of the same graph.

## beetles and support

Let C be a conjugacy class of the symmetric group Sn . Recall that C consists of all of the permutations of a given cycle type.

Hence we may define the support of C, denoted by supp(C), to be the number of nonfixed digits under the action of a permutation in C.

If C has “large” support, then each element of C“moves” a lot of letters.

If the permutations in C are all even, then we say that C is an even conjugacy class. If the permutations in C are all odd, we say that C is an odd conjugacy class.

For n≥5, the subgroup generated by C, denoted by C, is Sn if C is odd, and An if C is even.

now we understand how turing thought about avalanche -and the significance of 4. he used gccs terms, like beetles….

## From Enigma to rods to logs (to a fractional base)

The proper (fractional) base used when cryptanalyzing runny messages was not such a strange idea. After all, napiers original log tables were conceived in term of using a base that was a small offset from 1. E.g. O.9999999

I you were 1920 trained math person, you be trained in such “strange” log tables – and such mechanical devices with vertical wooden “rods” bearing abacus like beads. The latter could compute, using the log basis in question.

So think! Tunny back to rods.

And from rods we get back to enigma/Hebern crytanalysis, and isomorph searching.

Of course, rods in enigma algebra are less about logs and more about relative automorphisms (as one computes conjugates).

But you see the eureka transitions.

Once turing and co figured that a discriminator could exist for a rotor set, now one can leap to its log (and an algebra of bulges).

What unclassified docs don’t say is the parallel analysis going on with quantum mechanics calculations, given the parallel effort going on with the ;mostly compartmentalized) atomic bomb making efforts (circa 1944).

Must have been fun to be thinking about the ” potential” of the electron cloud in a colossus tube/valve and that similar controlled electron (well neuron) flow used to accelerate a uranium chain reaction.